Chapter 3 - Second Order Linear Equations - 3.1 Homogenous Equations with Constant Coefficients - Problems - Page 144: 3

$${y}=c_{1}\exp^{-\frac{1}{3}x}+c_{2}\exp^{\frac{1}{2}x}$$

$$6y''-y'-y=0$$Let $y-e^{\lambda{x}}$ so that $(\ln{y})'=\lambda$. $$6{\lambda}^2-{\lambda}-1=0$$ $$(3\lambda+1)(2\lambda-1)=0$$ $$\lambda_{1,2}=-\frac{1}{3},\frac{1}{2}$$ $$\therefore{y}=c_{1}\exp^{-\frac{1}{3}x}+c_{2}\exp^{\frac{1}{2}x}$$